In this paper, we address the analysis and state-feedback synthesis problems for linear parameter-varying (LPV) systems with parameter-varying time delays. It is assumed that the state-space data and the time delays depend on parameters that are measurable in real-time and vary in a compact set with bounded variation rates. We explore the delay-dependent stability and the induced L2 norm performance of these systems using parameter-dependent Lyapunov functions. In addition, the state-feedback control synthesis problem is examined when a variable state delay is present. Both analysis and synthesis conditions are formulated in terms of linear matrix inequalities (LMIs) that can be solved via efficient interior-point algorithms.